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Range Reporting for Time Series via Rectangle Stabbing

Authors Lotte Blank , Anne Driemel

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Author Details

Lotte Blank
  • University of Bonn, Germany
Anne Driemel
  • University of Bonn, Germany

Funding

This work was funded by 390685813 (Germany’s Excellence Strategy – EXC-2047/1: Hausdorff Center for Mathematics); 416767905; and the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) – 459420781 (FOR AlgoForGe).

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Lotte Blank and Anne Driemel. Range Reporting for Time Series via Rectangle Stabbing. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.15

Abstract

We study the Fréchet queries problem. It is a data structure problem for range reporting, where we are given a set S of n polygonal curves and a distance threshold ρ. The data structure should support queries with a polygonal curve q for the elements of S, for which the continuous Fréchet distance to q is at most ρ. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves of constant complexity and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point q should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with size in 𝒪(n log^{t-1} n) and query time in 𝒪(log^{t-1} n+k), where k denotes the output size and t can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. Note that we omit factors depending on the complexity of the curves that do not depend on n. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter d = ⌊t/2⌋ can be applied to our problem via reduction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Data Structures
  • Fréchet distance
  • Rectangle Stabbing
  • Orthogonal Range Searching

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References

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